I have a few questions to figure out Peskin 4.3 problem which is Linear sigma model about the interactions of pions at low energy. This model consist of N scalar fields governed by the Hamiltonian ($ i = 1,2,3....,N)$ $$ H=\int d^3x (\frac{1}{2}|\partial_{\mu}\Phi^i|^2+V(\Phi^2)) $$ $$ V(\Phi^2)=\frac{1}{2}m^2(\Phi^i)^2+\frac{\lambda}{4}(\Phi^i)^4 $$ I find the min/max of the potential energy $V(\Phi^2)$, by assing negative mass $$ \frac{\partial V}{\partial\Phi}=m^2\Phi+\lambda(\Phi^3)=0 $$ $$ m^2=-\mu^2 $$ $$ \Phi(-\mu^2+\lambda\Phi^2)=0 $$ $$ \Phi=\frac{\mu}{\sqrt{\lambda}}=v $$ It is the vacuum expectation value of the field and since it's not zero, it causes spontaneous symmetry breaking. In part b, Peskin assign N scalar field as $$ \Phi^i(x)=\pi^i(x) $$ $$ \Phi^N(x)=v+\sigma(x) $$ v is a constant $\pi^i$pion field and $i=1,2,3...,N-1$.
Shifted potential, after some algebra, is like below then $$ V(\Phi^2)=\mu^2\sigma^2+\frac{-\mu^4}{4\lambda}+\frac{\lambda}{4}\pi^4+\frac{\lambda}{2}\pi^2\sigma^2+\sqrt{\lambda}\mu\sigma\pi^2+\sqrt{\lambda}\mu\sigma^3+\frac{\lambda}{4}\sigma^4 $$ Here we see $\sigma$ field has a mass term. My question is, what is this field which gain mass and why pion fields become massless suddenly? First I assigned mass to pion fields , $-\mu$, but at the end I get massless pion fields and a field $\sigma$ with $\sqrt{2}\mu$ mass. Is it somehow about pions are the pseudo-Nambu-Goldstone bosons of spontaneously broken chiral symmetry as written wikipedia ?
